Question

Find the missing number:
$3,12,27,48,75,108,?$
A). $192$
B). $183$
C). $162$
D). $147$

Answer 1

Hint: In order to find the missing value from the series, initiate with finding the type of series that is being followed. For example, we can see that all the values are multiples of 3, so there must be some value increasing or decreasing in the products to get the next term. So, we need to find the terms.

Complete step-by-step solution:
We are given a series $3,12,27,48,75,108,?$. We need to find the value that is missing in the form of the question mark.
For that we are starting with finding the series that is being followed, and we can observe that there are multiples of 3.
So, we can write the second term with respect to 3 as:
$12 = 3 \times \left( 4 \right)$ …..(1)
Now, the next succeeding term as:
$27 = 3 \times \left( 9 \right)$ …..(2)
The difference between 9 and 4 from equation 1 and 2 is 5.
For the next term, we can write as:
$48 = 3 \times \left( {16} \right)$ …..(3)
The difference between 16 and 9 from equation 2 and 3 is 7.
For the next term, we can write as:
$75 = 3 \times \left( {25} \right)$ …..(4)
The difference between 25 and 16 from equation 3 and 4 is 9.
So, we can see that the value is increasing by two , so we can check by adding 2 to the value of 9 and adding it to 25 and them multiplying it with 3 to get the next term, so we get:
$9 + 2 = 11$
Adding 11 to 25, we get:
$11 + 25 = 36$
Multiplying 3 to 36:
$3 \times \left( {36} \right) = 108$
Which is the next term after 75, so the pattern is correct. So, adding 2 to 11 to continue the pattern:
$11 + 2 = 13$
Adding 13 to 36, we get:
$13 + 36 = 49$
Multiplying 3 to 49:
$3 \times \left( {49} \right) = 147$
Which would be the next term.
Therefore, the missing value would be $147$.
Hence, Option D is correct.

Note: It is not compulsory for a series to have only one pattern that is being followed. For example, if we observe in the series $3,12,27,48,75,108,?$. We can also follow the pattern as: $3 \times {1^2} = 3$
$3 \times {2^2} = 3 \times 4 = 12$
$3 \times {3^2} = 3 \times 9 = 27$
$3 \times {4^2} = 3 \times 16 = 48$
$3 \times {5^2} = 3 \times 25 = 75$
$3 \times {6^2} = 3 \times 36 = 108$
So, the next term will be:
$3 \times {7^2} = 3 \times 49 = 147$
In this way we can find the missing value in the series, and can follow any pattern.